Dirichlet problems for second order linear elliptic equations with $ L^{1} $-data

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $\Omega$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f \;\;\text{ in $\Omega$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial \Omega$} $$ and $$ - {\rm div} \left( A D u \right) + {\rm div}(ub) + cu = {\rm div} F \;\;\text{ in $\Omega$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial \Omega$} , $$ where $A=[a^{ij}]$ is symmetric, uniformly elliptic, and of vanishing mean oscillation (VMO). The main purposes of this paper is to study unique solvability for both problems with $L^1$-data. We prove that if $\Omega$ is of class $C^{1}$, $ {\rm div} A + b\in L^{n,1}(\Omega;\mathbb{R}^n)$, $c\in L^{\frac{n}{2},1}(\Omega) \cap L^s(\Omega)$ for some $1